In
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and related branches of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a totally disconnected space is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
that has only
singletons as
connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) are connected; in a totally disconnected space, these are the ''only'' connected proper subsets.
An important example of a totally disconnected space is the
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883.
T ...
, which is
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
to the set of
''p''-adic integers. Another example, playing a key role in
algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic o ...
, is the field of
''p''-adic numbers.
Definition
A topological space
is totally disconnected if the
connected components in
are the one-point sets. Analogously, a topological space
is totally path-disconnected if all
path-components in
are the one-point sets.
Another closely related notion is that of a
totally separated space, i.e. a space where
quasicomponents are singletons. That is, a topological space
is totally separated space if and only if for every
, the intersection of all
clopen neighborhoods of
is the singleton
. Equivalently, for each pair of distinct points
, there is a pair of disjoint open neighborhoods
of
such that
.
Every totally separated space is evidently totally disconnected but the converse is false even for
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
s. For instance, take
to be the Cantor's teepee, which is the
Knaster–Kuratowski fan
In topology, a branch of mathematics, the Knaster–Kuratowski fan (named after Polish mathematicians Bronisław Knaster and Kazimierz Kuratowski) is a specific connected space, connected topological space with the property that the removal o ...
with the apex removed. Then
is totally disconnected but its quasicomponents are not singletons. For
locally compact Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the ma ...
s the two notions (totally disconnected and totally separated) are equivalent.
Unfortunately in the literature (for instance
), totally disconnected spaces are sometimes called hereditarily disconnected, while the terminology totally disconnected is used for totally separated spaces.
Examples
The following are examples of totally disconnected spaces:
*
Discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest top ...
s
* The
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s
* The
irrational numbers
* The ''p''-adic numbers; more generally, all
profinite groups are totally disconnected.
* The
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883.
T ...
and the
Cantor space
* The
Baire space
* The
Sorgenfrey line
* Every Hausdorff space of
small inductive dimension
In the mathematical field of topology, the inductive dimension of a topological space ''X'' is either of two values, the small inductive dimension ind(''X'') or the large inductive dimension Ind(''X''). These are based on the observation that, in ...
0 is totally disconnected
* The
Erdős space
In mathematics, Erdős space is a topological space named after Paul Erdős, who described it in 1940. Erdős space is defined as a subspace E\subset\ell^2 of the Hilbert space of square summable sequences, consisting of the sequences whose ele ...
ℓ
''2'' is a totally disconnected Hausdorff space that does not have small inductive dimension 0.
*
Extremally disconnected Hausdorff spaces
*
Stone spaces
* The
Knaster–Kuratowski fan
In topology, a branch of mathematics, the Knaster–Kuratowski fan (named after Polish mathematicians Bronisław Knaster and Kazimierz Kuratowski) is a specific connected space, connected topological space with the property that the removal o ...
provides an example of a connected space, such that the removal of a single point produces a totally disconnected space.
Properties
*
Subspaces,
products, and
coproducts of totally disconnected spaces are totally disconnected.
*Totally disconnected spaces are
T1 spaces, since singletons are closed.
*Continuous images of totally disconnected spaces are not necessarily totally disconnected, in fact, every
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
is a continuous image of the
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883.
T ...
.
*A locally compact Hausdorff space has
small inductive dimension
In the mathematical field of topology, the inductive dimension of a topological space ''X'' is either of two values, the small inductive dimension ind(''X'') or the large inductive dimension Ind(''X''). These are based on the observation that, in ...
0 if and only if it is totally disconnected.
*Every totally disconnected compact metric space is homeomorphic to a subset of a
countable product of
discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest top ...
s.
*It is in general not true that every open set in a totally disconnected space is also closed.
*It is in general not true that the closure of every open set in a totally disconnected space is open, i.e. not every totally disconnected Hausdorff space is
extremally disconnected.
Constructing a totally disconnected quotient space of any given space
Let
be an arbitrary topological space. Let
if and only if
(where
denotes the largest connected subset containing
). This is obviously an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
whose equivalence classes are the connected components of
. Endow
with the
quotient topology, i.e. the
finest topology
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies.
Definition
A topology on a set may be defined as th ...
making the map
continuous. With a little bit of effort we can see that
is totally disconnected.
In fact this space is not only ''some'' totally disconnected quotient but in a certain sense the ''biggest'': The following
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
holds: For any totally disconnected space
and any continuous map
, there exists a ''unique'' continuous map
with
.
See also
*
Extremally disconnected space
*
Totally disconnected group In mathematics, a totally disconnected group is a topological group that is totally disconnected. Such topological groups are necessarily Hausdorff.
Interest centres on locally compact totally disconnected groups (variously referred to as group ...
References
* (reprint of the 1970 original, {{MR, 0264581)
General topology
Properties of topological spaces